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Consider G and H as simple graphs. A graph G has an H-covering if every edge of G belongs to a subgraph of G which the subgraph is isomorphic to H. The graph G is said to be H-magic if there exists a total labeling of G such that for every subgraph H′ of G where H′ is isomorphic to H, then the sum of all labels in H′ is constant. In this paper, we study grid-like graphs G, whose faces in F(G) are labeled. The magic labeling of type (1,1,1) means that the label is assigned to all vertices, edges, and faces, and the weight is calculated as ∑v∈V(G) l(v)+∑e∈E(G) l(e)+∑f∈F(G) l(f) with all weight for every face being constant. We show that certain unions of grid graphs are H-magic, where H is a grid graph. [ABSTRACT FROM AUTHOR]

A rainbow path in a graph G is an edge colored graph such that each edge in the path does not have the same color. The minimal number of colors which is needed to color the edges in the path so that every pair of vertices is connected by at least one rainbow path is called the rainbow connection number of G, denoted by rc(G). A rainbow path of length d(u, v), where d(u, v) is the distance between u and v, is called rainbow u - v geodesic in G. A strongly rainbow connected graph is a graph such that there exists a rainbow u − v geodesic for any two vertices u and v in G. The strong rainbow connection number src(G) of G is the minimum number of colors needed to make G strongly rainbow connected graph. In this paper, we prove the exact values of rc and src for crystal graph and its generalization to neuron graph. [ABSTRACT FROM AUTHOR]

Let F, G, and H be simple graphs. We say F → (G,H) if for every 2-coloring of the edges of F there exist a monochromatic G or H in F. The Ramsey number r(G,H) is min {v(F): F → (G,H)}, the size Ramsey number r* (G,H) is min {e(F) : F → (G,H)}, and the restricted size Ramsey number r* (G,H) is min {e(F) : F → (G,H), (F) = r(G,H)}. In 1972, Chvátal and Harary gave the Ramsey number for P3 versus any graph H with no isolates. In 1983, Harary and Miller started the investigation of the (restricted) size Ramsey number for some pairs of small graphs with order at most four. In1983, Faudree and Sheehan continued the investigation and summarized the complete results on the (restricted) size Ramsey number for all pairs of small graphs with order at most four. In 1998, Lortz and Mengenser gave both the size Ramsey number and the restricted size Ramsey number for all pairs of small forests with order at most five. Lately, we investigate the restricted size Ramsey number for a path of order three versus all connected graphs of order five. In this work, we continue the investigation on the restricted size Ramsey number for a pair of small graphs. In particularly, for a path of order three versus connected graphs of order six. [ABSTRACT FROM AUTHOR]